Consider a particle moving in a central potential. 
U = U(r),  F = f(r)(r/r),  f(r) = (-dU/dr).
In a central potential the energy E and the angular momentum M are conserved.  The motion is in a plane.  Choose this plane to be the x-y plane.  Then

.

.

Kepler’s second law:

.

The area swept out per unit time is constant for all central potentials.

Equations of motion involving r only:

,

yields

.

Lagrange’s equations yield

.

Equations for the orbit:

.

From

we obtain

,

,

.

From

we obtain

,

or

.

The Kepler problem

Let

.

The equation for the orbit yields

.

.

This is the equation of a conic section.  Here e is the eccentricity.

The ellipse and the circle are closed orbits.  For closed orbits we have

semi-major axis:

semi-minor axis:

From

we find the period

.

This is Kepler’s third law.

Two interacting particles

Consider two particles with masses m₁ and m₂, subject to internal forces
F₁₂ = F₁₂(r₂ - r₁)/r₁₂,    r₁₂ = |r₁ - r₂|,     F₂₁ = - F₁₂.
The equations of motion are 
m₁(dv₁/dt) = F₂₁,  m₂(dv₂/dt ) = F₁₂  = - F₂₁,  m₁dv₁/dt +  m₂dv₂/dt = 0.

Define the center of mass (CM) coordinate R = (m₁r₁ + m₂r₂)/(m₁+ m₂).
The center of mass moves with constant velocity.

Define the relative coordinate  r = r₁ - r₂.  
Then r₁ = R + (m₂/(m₁+m₂))r,  r₂  =  R - (m₁/(m₁ + m₂))r .
We find that (m₁m₂/(m₁+ m₂ ))d²r/dt² = F₂₁(r), or md²r/dt² = F₂₁(r). 
Here m = m₁m₂/(m₁ + m₂) is called the  reduced mass.
The problem of the relative motion of two interacting masses m₁and m₂ can be solved by solving for the motion of one fictitious particle of reduced mass m in a central field.

Equation of motion of the fictitious particle:       md²r/dt² = F₂₁(r) = F₂₁(r)(r/|r|).

The equations of the motion for the system can be separated into equations for the center of mass motion and equations for the relative motion.  Similarly, the sum of the momenta and sum of the angular momenta can be split into parts pertaining to the center of mass motion and parts pertaining to the relative motion.

In the CM frame of two interacting particles the Lagrangian

can be written as

.

The problem of two interacting particles in their CM frame is equivalent to the problem of a fictitious particle of reduced mass m moving in a central potential U(r).